and corresponding Gravitational Fields. 705 



where 



b' =T(r' r"-r' r" +£ r"-^-r"Y 



The latter equation expressos the necessary and sufficient 

 condition that, bv a suitable choice of coordinates, the square 

 of the element of length can be put in the form : — 



1,2, 3,4 



ds 2 = X d ** 2 (7) 



and the gravitational field made to vanish everywhere. Such 

 a field may conveniently be termed a non-permanent one. 

 The manifolds (2), (3), and (4) furnish simple illustrations of 

 such fields. 



In the manifold (2) the values of the g KT and gxr are : — 



F 0i i=t-1 Cg n = aH 2 -l 



1 9"' 1 



| W = 1-aV \g^= 1 



l#i4 =— «*> Li/ 14 = — «*, 



and 1 



= ~1 3 



where <j is the determinant of the g K r. 



The (Jliristoffel expressions T^ all vanish with the 



exception of T^ which has the value 



T' H = «. 

 The equations 



~V2 1» 2, 3, 4 -n -n 



O X<j ^ <t OV a 0.1'$ 



^ + ^^ b f=0, .= 1,2,3,4, . (8) 



a/3 



of the geodesies then give the following equations of 

 motion : — 



d 2 x _ 



as* 



*» = 



</*•- ' 

 Ji = o, 



